4 MFP Tetbook A-level Further Mathematics 660 Chapter : Comple Numbers Itroductio The geeral comple umber The modulus ad argumet of a comple umber 4 The polar form of a comple umber 5 Additio, subtractio ad multiplicatio of comple umbers of the form iy 6 The cojugate of a comple umber ad the divisio of comple umbers of the form iy 7 Products ad quotiets of comple umbers i their polar form 8 Equatig real ad imagiary parts 9 Further cosideratio of z z ad arg( z z) 0 Loci o Argad diagrams This chapter itroduces the idea of a comple umber Whe you have completed it, you will: kow what is meat by a comple umber; kow what is meat by the modulus ad argumet of a comple umber; kow how to add, subtract, multiply ad divide comple umbers; kow how to solve equatios usig real ad imagiary parts; uderstad what a Argad diagram is; kow how to sketch loci o Argad diagrams 4

5 MFP Tetbook A-level Further Mathematics 660 Itroductio You will have discovered by ow that some problems caot be solved i terms of real umbers For eample, if you use a calculator to evaluate 64 you get a error message This is because squarig every real umber gives a positive value; both ( 8) ad ( 8) are equal to 64 As caot be evaluated, a symbol is used to deote it the symbol used is i i i It follows that i The geeral comple umber The most geeral umber that ca be writte dow has the form i y, where ad y are real umbers The term iy is a comple umber with beig the real part ad y the imagiary part So, both i ad 4i are comple umbers The set of real umbers, (with which you are familiar), is really a subset of the set of comple umbers, This is because real umbers are actually umbers of the form 0i 5

6 MFP Tetbook A-level Further Mathematics 660 The modulus ad argumet of a comple umber Just as real umbers ca be represeted by poits o a umber lie, comple umbers ca be represeted by poits i a plae The poit P(, y) i the plae of coordiates with aes O ad Oy represets the comple umber iy ad the umber is uiquely represeted by that poit The diagram of poits i Cartesia coordiates represetig comple umbers is called a Argad diagram y r P(, y) O θ If the comple umber iy is deoted by z, ad hece z i, y z ( mod zed ) is defied as the distace from the origi O to the poit P represetig z Thus z OP r The modulus of a comple umber z is give by z y The argumet of z, arg z, is defied as the agle betwee the lie OP ad the positive -ais usually i the rage (π, π) The argumet of a comple umber z is give y by arg z, where ta You must be careful whe or y, or both, are egative 6

7 MFP Tetbook A-level Further Mathematics 660 Eample Fid the modulus ad argumet of the comple umber i Solutio The poit P represetig this umber, z, is show o the diagram y P, z ( ) ad ta θ Therefore, arg z π O Note that whe ta, θ could equal π or π However, the sketch clearly shows that θ lies i the secod quadrat This is why you eed to be careful whe evaluatig the argumet of a comple umber Eercise A Fid the modulus ad argumet of each of the followig comple umbers: (a) i, (b) i, (c) 4, (d) i Give your aswers for arg z i radias to two decimal places Fid the modulus ad argumet of each of the followig comple umbers: (a) i, (b) 4i, (c) 7i Give your aswers for arg z i radias to two decimal places 7

8 MFP Tetbook A-level Further Mathematics The polar form of a comple umber y I the diagram alogside, y rsi rcos ad P(, y) If P is the poit represetig the comple umber z i, y it follows that z may be writte i the form rcos irsi This is called the polar, or modulus argumet, form of a comple umber O θ r A comple umber may be writte i the form z r(cos isi ), where z r ad arg z For brevity, r(cos isi ) ca be writte as (r, θ) Eercise B Write the comple umbers give i Eercise A i polar coordiate form Fid, i the form i y, the comple umbers give i polar coordiate form by: (a) z cosπ π isi, (b) 4 4 π π 4cos isi 8

10 MFP Tetbook A-level Further Mathematics The cojugate of a comple umber ad the divisio of comple umbers of the form + iy The cojugate of a comple umber z iy (usually deoted by z * or z ) is the comple umber z* i y Thus, the cojugate of i is i, ad that of i is i O a Argad diagram, the poit represetig the comple umber z * is the reflectio of the poit represetig z i the -ais The most importat property of z * is that the product zz * is real sice zz* ( i y)( i y) iy iy i y y * zz z Divisio of two comple umbers demads a little more care tha their additio or multiplicatio ad usually requires the use of the comple cojugate Eample 6 z Simplify z, where z 4i ad z i Solutio 4i (4i)( i) i (i)(i) 4i6i8i i i 4i 50i 5 i z multiply the umerator ad deomiator of z by z *, ie ( i) so that the product of the deomiator becomes a real umber Eercise D z For the sets of comple umbers z ad z, fid z where (a) z 4 i ad z i, (b) z 6i ad z i 0

15 MFP Tetbook A-level Further Mathematics Loci o Argad diagrams A locus is a path traced out by a poit subjected to certai restrictios Paths ca be traced out by poits represetig variable comple umbers o a Argad diagram just as they ca i other coordiate systems Cosider the simplest case first, whe the poit P represets the comple umber z such that z k This meas that the distace of P from the origi O is costat ad so P will trace out a circle z k represets a circle with cetre O ad radius k If istead zz k, where z is a fied comple umber represeted by the poit A o a Argad diagram, the (from Sectio 9) z z represets the distace AP ad is costat It follows that P must lie o a circle with cetre A ad radius k zz k represets a circle with cetre z ad radius k Note that if zz k, the the poit P represetig z ca ot oly lie o the circumferece of the circle, but also aywhere iside the circle The locus of P is therefore the regio o ad withi the circle with cetre A ad radius k Now cosider the locus of a poit P represeted by the comple umber z subject to the coditios zz z z, where z ad z are fied comple umbers represeted by the poits A ad B o a Argad diagram Agai, usig the result of Sectio 9, it follows that AP BP because z z is the distace AP ad z z is the distace BP Hece, the locus of P is a straight lie zz z z represets a straight lie the perpedicular bisector of the lie joiig the poits z ad z Note also that if zz z z the locus of z is ot oly the perpedicular bisector of AB, but also the whole half plae, i which A lies, bouded by this bisector All the loci cosidered so far have bee related to distaces there are also simple loci i Argad diagrams ivolvig agles The simplest case is the locus of P subject to the coditio that arg z, where is a fied agle y O α P 5

16 MFP Tetbook A-level Further Mathematics 660 This coditio implies that the agle betwee OP ad O is fied ( ) so that the locus of P is a straight lie arg z represets the half lie through O iclied at a agle to the positive directio of O Note that the locus of P is oly a half lie the other half lie, show dotted i the diagram above, would have the equatio arg z π, possibly π if π falls outside the specified rage for arg z I eactly the same way as before, the locus of a poit P satisfyig arg( zz), where z is a fied comple umber represeted by the poit A, is a lie through A arg( z z ) iclied at a agle to the positive directio of O represets the half lie through the poit z y P A α O Note agai that this locus is oly a half lie the other half lie would have the equatio arg( zz ) π, possibly π Fially, cosider the locus of ay poit P satisfyig arg( zz) This idicates that the agle betwee AP ad the positive -ais lies betwee ad, so that P ca lie o or withi the two half lies as show shaded i the diagram below y A β α O 6

20 MFP Tetbook A-level Further Mathematics (a) Idicate o a Argad diagram the regio of the comple plae i which 0arg π z (b) The comple umber z is such that 0arg π z ad π arg z π 6 (i) Sketch aother Argad diagram showig the regio R i which z must lie (ii) Mark o this diagram the poit A belogig to R at which z has its least possible value (c) At the poit A defied i part (b)(ii), z z A (i) Calculate the value of z A (ii) Epress z A i the form a i b [AQA March 999] 9 (a) The comple umbers z ad w are such that z 4i i ad w 4 i i Epress each of z ad w i the form a i b, where a ad b are real (b) (i) Write dow the modulus ad argumet of each of the comple umbers 4 i ad i Give each modulus i a eact surd form ad each argumet i radias betwee π ad π (ii) The poits O, P ad Q i the comple plae represet the comple umbers 0 0i, 4 i ad i, respectively Fid the eact legth of PQ ad hece, or otherwise, show that the triagle OPQ is right-agled [AEB Jue 997] 0

21 MFP Tetbook A-level Further Mathematics 660 Chapter : Roots of Polyomial Equatios Itroductio Quadratic equatios Cubic equatios 4 Relatioship betwee the roots of a cubic equatio ad its coefficiets 5 Cubic equatios with related roots 6 A importat result 7 Polyomial equatios of degree 8 Comple roots of polyomial equatios with real coefficiets This chapter revises work already covered o roots of equatios ad eteds those ideas Whe you have completed it, you will: kow how to solve ay quadratic equatio; kow that there is a relatioship betwee the umber of real roots ad form of a polyomial equatio, ad be able to sketch graphs; kow the relatioship betwee the roots of a cubic equatio ad its coefficiets; be able to form cubic equatios with related roots; kow how to eted these results to polyomials of higher degree; kow that comple cojugates are roots of polyomials with real coefficiets

22 MFP Tetbook A-level Further Mathematics 660 Itroductio You should have already met the idea of a polyomial equatio A polyomial equatio of degree, oe with as the highest power of, is called a quadratic equatio Similarly, a polyomial equatio of degree has as the highest power of ad is called a cubic equatio; oe with 4 as the highest power of is called a quartic equatio I this chapter you are goig to study the properties of the roots of these equatios ad ivestigate methods of solvig them Quadratic equatios You should be familiar with quadratic equatios ad their properties from your earlier studies of pure mathematics However, eve if this sectio is familiar to you it provides a suitable base from which to move o to equatios of higher degree You will kow, for eample, that quadratic equatios of the type you have met have two roots (which may be coicidet) There are ormally two ways of solvig a quadratic equatio by factorizig ad, i cases where this is impossible, by the quadratic formula Graphically, the roots of the equatio a b c 0 are the poits of itersectio of the curve y a b c ad the lie y 0 (ie the -ais) For eample, a sketch of part of y 8 is show below y ( 4, 0) (, 0) (0, 8) The roots of this quadratic equatio are those of ( )( 4) 0, which are ad 4

23 MFP Tetbook A-level Further Mathematics 660 A sketch of part of the curve y 4 4 is show below y (0, 4) O (, 0) I this case, the curve touches the -ais The equatio ( ) 0 ad, a repeated root 44 0 may be writte as Not all quadratic equatios are as straightforward as the oes cosidered so far A sketch of part of the curve y 4 5 is show below y (0, 5) O (, ) This curve does ot touch the -ais so the equatio 45 0 caot have real roots Certaily, 4 5 will ot factorize so the quadratic formula b b 4ac a has to be used to solve this equatio This leads to 4 60 ad, usig ideas from Chapter, this becomes 4 i or i It follows that the equatio 45 0 does have two roots, but they are both comple umbers I fact the two roots are comple cojugates You may also have observed that whether a quadratic equatio has real or comple roots depeds o the value of the discrimiat b 4 ac The quadratic equatio a b c 0, where a, b ad c are real, has comple roots if b 4ac 0

24 MFP Tetbook A-level Further Mathematics 660 Eercise A Solve the equatios (a) 60 0, (b) 06 0 Cubic equatios As metioed i the itroductio to this chapter, equatios of the form a b c d 0 are called cubic equatios All cubic equatios have at least oe real root ad this real root is ot always easy to locate The reaso for this is that cubic curves are cotiuous they do ot have asymptotes or ay other form of discotiuity Also, as, the term a becomes the domiat part of the epressio ad a (if a 0), whilst a whe Hece the curve must cross the lie y 0 at least oce If a 0, the as, ad a as ad this does ot affect the result a 4

25 MFP Tetbook A-level Further Mathematics 660 A typical cubic equatio, below y y a b c d with a 0, ca look like ay of the sketches The equatio of this curve has three real roots because the curve crosses the lie y 0 at three poits O y y O O I each of the two sketch graphs above, the curve crosses the lie y 0 just oce, idicatig just oe real root I both cases, the cubic equatio will have two comple roots as well as the sigle real root Eample (a) Fid the roots of the cubic equatio 0 (b) Sketch a graph of y Solutio y 4 (a) If f( ), the f() 0 Therefore is a factor of f() f( ) ( )( 4) ( )( )( ) Hece the roots of f() = 0 are, ad (b)

27 MFP Tetbook A-level Further Mathematics Relatioship betwee the roots of a cubic equatio ad its coefficiets As a cubic equatio has three roots, which may be real or comple, it follows that if the geeral cubic equatio a b c d 0 has roots, ad, it may be writte as a ( )( )( ) 0 Note that the factor a is required to esure that the coefficiets of are the same, so makig the equatios idetical Thus, o epadig the right had side of the idetity, a b c d a( )( )( ) a a( ) a( ) a The two sides are idetical so the coefficiets of ad ca be compared, ad also the umber terms, ba( ) ca( ) d a If the cubic equatio a b c d 0 has roots, ad, the b, a c, a d a Note that meas the sum of all the roots, ad that meas the sum of all the possible products of roots take two at a time Eercise C Fid, ad for the followig cubic equatios: (a) 7 5 0, (b) The roots of a cubic equatio are, ad If, the cubic equatio 7 ad 5, state 7

29 MFP Tetbook A-level Further Mathematics 660 (c) So that the cubic equatio with roots, ad is or 4 0 9

30 MFP Tetbook A-level Further Mathematics 660 The secod method of fidig the cubic equatios i Eample 5 is show below It is ot always possible to use this secod method, but whe you ca it is much quicker tha the first Solutio: method (a) As the roots are to be, ad, it follows that, if X, the a cubic equatio i X must have roots which are twice the roots of the cubic equatio i As the equatio i is 4 0, if you substitute X the equatio i X becomes as before or X X 4 0, X 6X 0 (b) I this case, if you put X i 4 0, the ay root of a equatio i X must be less tha the correspodig root of the cubic i Now, X gives X ad substitutig ito 4 0 gives ( X ) ( X ) 4 0 which reduces to X X 0 (c) I this case you use the substitutio X or X For 4 0 X X O multiplyig by X, this gives X 4X 0 or 4X X 0 as before 4 0 this gives Eercise D The cubic equatio 47 0 has roots, ad Usig the first method described above, fid the cubic equatios whose roots are (a), ad, (b), ad, (c), ad Repeat Questio above usig the secod method described above Repeat Questios ad above for the cubic equatio 6 0 0

31 MFP Tetbook A-level Further Mathematics A importat result If you square you get ( ) ( )( ) So,, or for three umbers, ad This result is well worth rememberig it is frequetly eeded i questios ivolvig the symmetric properties of roots of a cubic equatio Eample 6 The cubic equatio has roots, ad Fid the cubic equatios with (a) roots, ad, (b) roots, ad [Note that the direct approach illustrated below is the most straightforward way of solvig this type of problem] Solutio (a) ( ) Hece the cubic equatio is (b) 5 6 usig the same result but replacig with with, ad with Thus 6 (5) 46 ( ) Hece the cubic equatio is 46 0

32 MFP Tetbook A-level Further Mathematics Polyomial equatios of degree The ideas covered so far o quadratic ad cubic equatios ca be eteded to equatios of ay degree A equatio of degree has two roots, oe of degree has three roots so a equatio of degree has roots Suppose the equatio a b c d k 0 has roots,,, the b, a c, a d, a ( ) k util, fially, the product of the roots a Remember that is the sum of the products of all possible pairs of roots, is the sum of the products of all possible combiatios of roots take three at a time, ad so o I practice, you are ulikely to meet equatios of degree higher tha 4 so this sectio cocludes with a eample usig a quartic equatio Eample 7 4 The quartic equatio has roots,, ad Write dow (a), (b) (c) Hece fid Solutio (a) (b) (c) 4 6 ( Now ( This shows that the importat result i Sectio 6 ca be eteded to ay umber of letters Hece ( ) ( ) 0

33 MFP Tetbook A-level Further Mathematics 660 Eercise E 4 The quartic equatio 58 0 has roots,, ad (a) Fid the equatio with roots,, ad (b) Fid 8 Comple roots of polyomial equatios with real coefficiets Cosider the polyomial equatio f( ) a b c k Usig the ideas from Chapter, if p ad q are real, f( pi q) a( pi q) b( pi q) k ui v, where u ad v are real Now, f( i ) ( i ) ( i ) p q a p q b p q k uiv sice i raised to a eve power is real ad is the same as i raised to a eve power, makig the real part of f ( p i q) the same as the real part of f ( p i q) But i raised to a odd power is the same as i raised to a odd power multiplied by, ad odd powers of i comprise the imagiary part of f ( p i q) Thus, the imagiary part of f ( p i q) is times the imagiary part of f ( p i q) Now if p iq is a root of f ( ) 0, it follows that u iv 0 ad so u 0 ad v 0 Hece, uiv 0 makig f ( p i q) 0 ad p iq a root of f ( ) 0 If a polyomial equatio has real coefficiets ad if p i q, where p ad q are real, is a root of the polyomial, the its comple cojugate, p i q, is also a root of the equatio It is very importat to ote that the coefficiets i f ( ) 0 must be real If f ( ) 0 has comple coefficiets, this result does ot apply

34 MFP Tetbook A-level Further Mathematics 660 Eample 8 The cubic equatio k 0, where k is real, has oe root equal to i Fid the other two roots ad the value of k Solutio As the coefficiets of the cubic equatio are real, it follows that i is also a root Cosiderig the sum of the roots of the equatio, if is the third root, ( i) ( i), To fid k, k ( i)( i)( ) 5, k 5 Eample 8 The quartic equatio roots has oe root equal to i Fid the other three Solutio As the coefficiets of the quartic are real, it follows that i is also a root Hece (i) ( i) is a quadratic factor of the quartic Now, (i) (i) (i) (i) (i)(i) 5 4 Hece 5 is a factor of Therefore 45 ( 5)( a b) Comparig the coefficiets of, a a 4 Cosiderig the umber terms, 5b 5 b Hece the quartic equatio may be writte as ( 5)( 4) 0 ( 5)( )( ) 0, ad the four roots are i,i, ad 4

35 MFP Tetbook A-level Further Mathematics 660 Eercise F A cubic equatio has real coefficiets Oe root is ad aother is i Fid the cubic equatio i the form a bc 0 The cubic equatio roots has oe root equal to i Fid the other two The quartic equatio three roots has oe root equal to i Fid the other 5

36 MFP Tetbook A-level Further Mathematics 660 Miscellaeous eercises The equatio 4 0, p where p is a costat, has roots, ad, where 0 (a) Fid the values of ad (b) Fid the value of p [NEAB Jue 998] The umbers, ad satisfy the equatios ad (a) Show that 0 (b) The umbers, ad are also the roots of the equatio 0, p q r where p, q ad r are real (i) Give that 4i ad that is real, obtai ad (ii) Calculate the product of the three roots (iii) Write dow, or determie, the values of p, q ad r [AQA Jue 000] The roots of the cubic equatio are, ad 4 0 (a) Write dow the values of, ad (b) Fid the cubic equatio, with iteger coefficiets, havig roots, ad [AQA March 000] 4 The roots of the equatio are, ad (a) Write dow the value of (b) Give that i is a root of the equatio, fid the other two roots [AQA Specime] 6

37 MFP Tetbook A-level Further Mathematics The roots of the cubic equatio 0, p q r where p, q ad r are real, are, ad (a) Give that, write dow the value of p (b) Give also that 5, (i) fid the value of q, (ii) eplai why the equatio must have two o-real roots ad oe real root (c) Oe of the two o-real roots of the cubic equatio is 4i (i) Fid the real root (ii) Fid the value of r [AQA March 999] 6 (a) Prove that whe a polyomial (b) The polyomial g is defied by 5 f is divided by a, g 6 p q, where p ad q are real costats Whe remaider is (i) Fid the values of p ad q [AQA Jue 999] (ii) Show that whe the remaider is a f g is divided by i, where i, the g is divided by i, the remaider is 6i 7

38 MFP Tetbook A-level Further Mathematics 660 Chapter : Summatio of Fiite Series Itroductio Summatio of series by the method of differeces Summatio of series by the method of iductio 4 Proof by iductio eteded to other areas of mathematics This chapter eteds the idea of summatio of simple series, with which you are familiar from earlier studies, to other kids of series Whe you have completed it, you will: kow ew methods of summig series; kow which method is appropriate for the summatio of a particular series; uderstad a importat method kow as the method of iductio; be able to apply the method of iductio i circumstaces other tha i the summatio of series 8

Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

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Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig

Hadout: How to calculate time complexity? CSE 101 Witer 014 Recipe (a) Kow algorithm If you are usig a modied versio of a kow algorithm, you ca piggyback your aalysis o the complexity of the origial algorithm

GCE A Level H Maths Solutio Paper SECTION A (PURE MATHEMATICS) (i) Im 3 Note: Uless required i the questio, it would be sufficiet to just idicate the cetre ad radius of the circle i such a locus drawig.

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Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σ-algebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio

Itroductio to fiite fields II Fiite field of p elemets F Because we are iterested i doig computer thigs it would be useful for us to costruct fields havig elemets. Let s costruct a field of elemets; we